tpr2rico - Mathbox for Thierry Arnoux

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Theorem tpr2rico 31155

Description: For any point of an open set of the usual topology on (ℝ × ℝ) there is an open square which contains that point and is entirely in the open set. This is square is actually a ball by the (𝑙↑+∞) norm 𝑋. (Contributed by Thierry Arnoux, 21-Sep-2017.)

**Hypotheses**
Ref	Expression
tpr2rico.0	⊢ 𝐽 = (topGen‘ran (,))
tpr2rico.1	⊢ 𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣)))
tpr2rico.2	⊢ 𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))

**Assertion**
Ref	Expression
tpr2rico	⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑟 ∈ 𝐵 (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴))

Distinct variable groups: 𝑣,𝑢,𝑥,𝑦 𝑥,𝑟,𝐴 𝐵,𝑟 𝑥,𝐺 𝑥,𝐽 𝑥,𝑋 𝑦,𝑟,𝑋 Allowed substitution hints: 𝐴(𝑦,𝑣,𝑢) 𝐵(𝑥,𝑦,𝑣,𝑢) 𝐺(𝑦,𝑣,𝑢,𝑟) 𝐽(𝑦,𝑣,𝑢,𝑟) 𝑋(𝑣,𝑢)

Proof of Theorem tpr2rico Dummy variables 𝑧 𝑚 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ioo 12743	. . . . . . . . . 10 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
2	1	ixxf 12749	. . . . . . . . 9 ⊢ (,):(ℝ^* × ℝ^)⟶𝒫 ℝ^
3		ffn 6514	. . . . . . . . 9 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ^ → (,) Fn (ℝ^* × ℝ^*))
4	2, 3	mp1i 13	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (,) Fn (ℝ^* × ℝ^*))
5		elssuni 4868	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → 𝐴 ⊆ ∪ (𝐽 ×_t 𝐽))
6		tpr2rico.0	. . . . . . . . . . . . . . . 16 ⊢ 𝐽 = (topGen‘ran (,))
7		retop 23370	. . . . . . . . . . . . . . . 16 ⊢ (topGen‘ran (,)) ∈ Top
8	6, 7	eqeltri 2909	. . . . . . . . . . . . . . 15 ⊢ 𝐽 ∈ Top
9		uniretop 23371	. . . . . . . . . . . . . . . 16 ⊢ ℝ = ∪ (topGen‘ran (,))
10	6	unieqi 4851	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,))
11	9, 10	eqtr4i 2847	. . . . . . . . . . . . . . 15 ⊢ ℝ = ∪ 𝐽
12	8, 8, 11, 11	txunii 22201	. . . . . . . . . . . . . 14 ⊢ (ℝ × ℝ) = ∪ (𝐽 ×_t 𝐽)
13	5, 12	sseqtrrdi 4018	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → 𝐴 ⊆ (ℝ × ℝ))
14	13	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝐴 ⊆ (ℝ × ℝ))
15		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑋 ∈ 𝐴)
16	14, 15	sseldd 3968	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑋 ∈ (ℝ × ℝ))
17		xp1st 7721	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (ℝ × ℝ) → (1^st ‘𝑋) ∈ ℝ)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (1^st ‘𝑋) ∈ ℝ)
19		simpr 487	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ⁺)
20	19	rpred 12432	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑑 ∈ ℝ)
21	20	rehalfcld 11885	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (𝑑 / 2) ∈ ℝ)
22	18, 21	resubcld 11068	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) − (𝑑 / 2)) ∈ ℝ)
23	22	rexrd 10691	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) − (𝑑 / 2)) ∈ ℝ^*)
24	18, 21	readdcld 10670	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) + (𝑑 / 2)) ∈ ℝ)
25	24	rexrd 10691	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) + (𝑑 / 2)) ∈ ℝ^*)
26		fnovrn 7323	. . . . . . . 8 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ ((1^st ‘𝑋) − (𝑑 / 2)) ∈ ℝ^ ∧ ((1^st ‘𝑋) + (𝑑 / 2)) ∈ ℝ^*) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ∈ ran (,))
27	4, 23, 25, 26	syl3anc 1367	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ∈ ran (,))
28		xp2nd 7722	. . . . . . . . . . 11 ⊢ (𝑋 ∈ (ℝ × ℝ) → (2^nd ‘𝑋) ∈ ℝ)
29	16, 28	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (2^nd ‘𝑋) ∈ ℝ)
30	29, 21	resubcld 11068	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) − (𝑑 / 2)) ∈ ℝ)
31	30	rexrd 10691	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) − (𝑑 / 2)) ∈ ℝ^*)
32	29, 21	readdcld 10670	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) + (𝑑 / 2)) ∈ ℝ)
33	32	rexrd 10691	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) + (𝑑 / 2)) ∈ ℝ^*)
34		fnovrn 7323	. . . . . . . 8 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ ((2^nd ‘𝑋) − (𝑑 / 2)) ∈ ℝ^ ∧ ((2^nd ‘𝑋) + (𝑑 / 2)) ∈ ℝ^*) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ∈ ran (,))
35	4, 31, 33, 34	syl3anc 1367	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ∈ ran (,))
36		eqidd 2822	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))))
37		xpeq1 5569	. . . . . . . . 9 ⊢ (𝑥 = (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) → (𝑥 × 𝑦) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × 𝑦))
38	37	eqeq2d 2832	. . . . . . . 8 ⊢ (𝑥 = (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) → (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦) ↔ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × 𝑦)))
39		xpeq2 5576	. . . . . . . . 9 ⊢ (𝑦 = (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × 𝑦) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))))
40	39	eqeq2d 2832	. . . . . . . 8 ⊢ (𝑦 = (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) → (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × 𝑦) ↔ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))))
41	38, 40	rspc2ev 3635	. . . . . . 7 ⊢ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ∈ ran (,) ∧ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ∈ ran (,) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))) → ∃𝑥 ∈ ran (,)∃𝑦 ∈ ran (,)((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦))
42	27, 35, 36, 41	syl3anc 1367	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ∃𝑥 ∈ ran (,)∃𝑦 ∈ ran (,)((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦))
43		eqid 2821	. . . . . . 7 ⊢ (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) = (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
44		vex 3497	. . . . . . . 8 ⊢ 𝑥 ∈ V
45		vex 3497	. . . . . . . 8 ⊢ 𝑦 ∈ V
46	44, 45	xpex 7476	. . . . . . 7 ⊢ (𝑥 × 𝑦) ∈ V
47	43, 46	elrnmpo 7287	. . . . . 6 ⊢ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ ran (,)∃𝑦 ∈ ran (,)((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) = (𝑥 × 𝑦))
48	42, 47	sylibr 236	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦)))
49		tpr2rico.2	. . . . 5 ⊢ 𝐵 = ran (𝑥 ∈ ran (,), 𝑦 ∈ ran (,) ↦ (𝑥 × 𝑦))
50	48, 49	eleqtrrdi 2924	. . . 4 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵)
51	50	ralrimiva 3182	. . 3 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∀𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵)
52		xpss 5571	. . . . . . 7 ⊢ (ℝ × ℝ) ⊆ (V × V)
53	52, 16	sseldi 3965	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑋 ∈ (V × V))
54	18	rexrd 10691	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (1^st ‘𝑋) ∈ ℝ^*)
55	19	rphalfcld 12444	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (𝑑 / 2) ∈ ℝ⁺)
56	18, 55	ltsubrpd 12464	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) − (𝑑 / 2)) < (1^st ‘𝑋))
57	18, 55	ltaddrpd 12465	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (1^st ‘𝑋) < ((1^st ‘𝑋) + (𝑑 / 2)))
58		elioo1 12779	. . . . . . . . 9 ⊢ ((((1^st ‘𝑋) − (𝑑 / 2)) ∈ ℝ^* ∧ ((1^st ‘𝑋) + (𝑑 / 2)) ∈ ℝ^) → ((1^st ‘𝑋) ∈ (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ↔ ((1^st ‘𝑋) ∈ ℝ^ ∧ ((1^st ‘𝑋) − (𝑑 / 2)) < (1^st ‘𝑋) ∧ (1^st ‘𝑋) < ((1^st ‘𝑋) + (𝑑 / 2)))))
59	23, 25, 58	syl2anc 586	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) ∈ (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ↔ ((1^st ‘𝑋) ∈ ℝ^* ∧ ((1^st ‘𝑋) − (𝑑 / 2)) < (1^st ‘𝑋) ∧ (1^st ‘𝑋) < ((1^st ‘𝑋) + (𝑑 / 2)))))
60	54, 56, 57, 59	mpbir3and 1338	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (1^st ‘𝑋) ∈ (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))))
61	29	rexrd 10691	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (2^nd ‘𝑋) ∈ ℝ^*)
62	29, 55	ltsubrpd 12464	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) − (𝑑 / 2)) < (2^nd ‘𝑋))
63	29, 55	ltaddrpd 12465	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (2^nd ‘𝑋) < ((2^nd ‘𝑋) + (𝑑 / 2)))
64		elioo1 12779	. . . . . . . . 9 ⊢ ((((2^nd ‘𝑋) − (𝑑 / 2)) ∈ ℝ^* ∧ ((2^nd ‘𝑋) + (𝑑 / 2)) ∈ ℝ^) → ((2^nd ‘𝑋) ∈ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ↔ ((2^nd ‘𝑋) ∈ ℝ^ ∧ ((2^nd ‘𝑋) − (𝑑 / 2)) < (2^nd ‘𝑋) ∧ (2^nd ‘𝑋) < ((2^nd ‘𝑋) + (𝑑 / 2)))))
65	31, 33, 64	syl2anc 586	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) ∈ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ↔ ((2^nd ‘𝑋) ∈ ℝ^* ∧ ((2^nd ‘𝑋) − (𝑑 / 2)) < (2^nd ‘𝑋) ∧ (2^nd ‘𝑋) < ((2^nd ‘𝑋) + (𝑑 / 2)))))
66	61, 62, 63, 65	mpbir3and 1338	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (2^nd ‘𝑋) ∈ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))
67	60, 66	jca 514	. . . . . 6 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) ∈ (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ∧ (2^nd ‘𝑋) ∈ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))))
68		elxp7 7724	. . . . . 6 ⊢ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ↔ (𝑋 ∈ (V × V) ∧ ((1^st ‘𝑋) ∈ (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ∧ (2^nd ‘𝑋) ∈ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))))
69	53, 67, 68	sylanbrc 585	. . . . 5 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))))
70	69	ralrimiva 3182	. . . 4 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∀𝑑 ∈ ℝ⁺ 𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))))
71		mnfle 12530	. . . . . . . . . . . . . . . . . 18 ⊢ (((1^st ‘𝑋) − (𝑑 / 2)) ∈ ℝ^* → -∞ ≤ ((1^st ‘𝑋) − (𝑑 / 2)))
72	23, 71	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → -∞ ≤ ((1^st ‘𝑋) − (𝑑 / 2)))
73		pnfge 12526	. . . . . . . . . . . . . . . . . 18 ⊢ (((1^st ‘𝑋) + (𝑑 / 2)) ∈ ℝ^* → ((1^st ‘𝑋) + (𝑑 / 2)) ≤ +∞)
74	25, 73	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((1^st ‘𝑋) + (𝑑 / 2)) ≤ +∞)
75		mnfxr 10698	. . . . . . . . . . . . . . . . . 18 ⊢ -∞ ∈ ℝ^*
76		pnfxr 10695	. . . . . . . . . . . . . . . . . 18 ⊢ +∞ ∈ ℝ^*
77		ioossioo 12830	. . . . . . . . . . . . . . . . . 18 ⊢ (((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ ≤ ((1^st ‘𝑋) − (𝑑 / 2)) ∧ ((1^st ‘𝑋) + (𝑑 / 2)) ≤ +∞)) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
78	75, 76, 77	mpanl12 700	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ≤ ((1^st ‘𝑋) − (𝑑 / 2)) ∧ ((1^st ‘𝑋) + (𝑑 / 2)) ≤ +∞) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
79	72, 74, 78	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
80		ioomax 12812	. . . . . . . . . . . . . . . 16 ⊢ (-∞(,)+∞) = ℝ
81	79, 80	sseqtrdi 4017	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ⊆ ℝ)
82		mnfle 12530	. . . . . . . . . . . . . . . . . 18 ⊢ (((2^nd ‘𝑋) − (𝑑 / 2)) ∈ ℝ^* → -∞ ≤ ((2^nd ‘𝑋) − (𝑑 / 2)))
83	31, 82	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → -∞ ≤ ((2^nd ‘𝑋) − (𝑑 / 2)))
84		pnfge 12526	. . . . . . . . . . . . . . . . . 18 ⊢ (((2^nd ‘𝑋) + (𝑑 / 2)) ∈ ℝ^* → ((2^nd ‘𝑋) + (𝑑 / 2)) ≤ +∞)
85	33, 84	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((2^nd ‘𝑋) + (𝑑 / 2)) ≤ +∞)
86		ioossioo 12830	. . . . . . . . . . . . . . . . . 18 ⊢ (((-∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ ≤ ((2^nd ‘𝑋) − (𝑑 / 2)) ∧ ((2^nd ‘𝑋) + (𝑑 / 2)) ≤ +∞)) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
87	75, 76, 86	mpanl12 700	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ≤ ((2^nd ‘𝑋) − (𝑑 / 2)) ∧ ((2^nd ‘𝑋) + (𝑑 / 2)) ≤ +∞) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
88	83, 85, 87	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ⊆ (-∞(,)+∞))
89	88, 80	sseqtrdi 4017	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ⊆ ℝ)
90		xpss12 5570	. . . . . . . . . . . . . . 15 ⊢ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) ⊆ ℝ ∧ (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))) ⊆ ℝ) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (ℝ × ℝ))
91	81, 89, 90	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (ℝ × ℝ))
92	91	sselda 3967	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))) → 𝑥 ∈ (ℝ × ℝ))
93	92	expcom 416	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝑥 ∈ (ℝ × ℝ)))
94	93	ancld 553	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ))))
95	94	imdistanri 572	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))) → ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ 𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))))
96	13	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ⁺ ∧ 𝑥 ∈ (ℝ × ℝ))) → 𝐴 ⊆ (ℝ × ℝ))
97		simpr1 1190	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ⁺ ∧ 𝑥 ∈ (ℝ × ℝ))) → 𝑋 ∈ 𝐴)
98	96, 97	sseldd 3968	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ (𝑋 ∈ 𝐴 ∧ 𝑑 ∈ ℝ⁺ ∧ 𝑥 ∈ (ℝ × ℝ))) → 𝑋 ∈ (ℝ × ℝ))
99	98	3anassrs 1356	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑋 ∈ (ℝ × ℝ))
100		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑥 ∈ (ℝ × ℝ))
101		simplr 767	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ⁺)
102	101	rphalfcld 12444	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑑 / 2) ∈ ℝ⁺)
103		tpr2rico.1	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (𝑢 ∈ ℝ, 𝑣 ∈ ℝ ↦ (𝑢 + (i · 𝑣)))
104	103	cnre2csqima 31154	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ (ℝ × ℝ) ∧ 𝑥 ∈ (ℝ × ℝ) ∧ (𝑑 / 2) ∈ ℝ⁺) → (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → ((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2))))
105	99, 100, 102, 104	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → ((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2))))
106		eqid 2821	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
107	103, 6, 106	cnrehmeo 23557	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐺 ∈ ((𝐽 ×_t 𝐽)Homeo(TopOpen‘ℂ_fld))
108	106	cnfldtopon 23391	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
109	108	toponunii 21524	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
110	12, 109	hmeof1o 22372	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺 ∈ ((𝐽 ×_t 𝐽)Homeo(TopOpen‘ℂ_fld)) → 𝐺:(ℝ × ℝ)–1-1-onto→ℂ)
111		f1of 6615	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → 𝐺:(ℝ × ℝ)⟶ℂ)
112	107, 110, 111	mp2b 10	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐺:(ℝ × ℝ)⟶ℂ
113	112	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝐺:(ℝ × ℝ)⟶ℂ)
114	113, 99	ffvelrnd 6852	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘𝑋) ∈ ℂ)
115	112	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → 𝐺:(ℝ × ℝ)⟶ℂ)
116	115	ffvelrnda 6851	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘𝑥) ∈ ℂ)
117		sqsscirc2 31152	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺‘𝑋) ∈ ℂ ∧ (𝐺‘𝑥) ∈ ℂ) ∧ 𝑑 ∈ ℝ⁺) → (((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2)) → (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑))
118	114, 116, 101, 117	syl21anc 835	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2)) → (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑))
119	118	imp 409	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ ((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2))) → (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑)
120	101	rpxrd 12433	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝑑 ∈ ℝ^*)
121	120	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → 𝑑 ∈ ℝ^*)
122		cnxmet 23381	. . . . . . . . . . . . . . . . 17 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
123	121, 122	jctil 522	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → ((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ^*))
124	114	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → (𝐺‘𝑋) ∈ ℂ)
125	116	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → (𝐺‘𝑥) ∈ ℂ)
126	124, 125	jca 514	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → ((𝐺‘𝑋) ∈ ℂ ∧ (𝐺‘𝑥) ∈ ℂ))
127		eqid 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ (abs ∘ − ) = (abs ∘ − )
128	127	cnmetdval 23379	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺‘𝑥) ∈ ℂ ∧ (𝐺‘𝑋) ∈ ℂ) → ((𝐺‘𝑥)(abs ∘ − )(𝐺‘𝑋)) = (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))))
129	125, 124, 128	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → ((𝐺‘𝑥)(abs ∘ − )(𝐺‘𝑋)) = (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))))
130		simpr 487	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑)
131	129, 130	eqbrtrd 5088	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → ((𝐺‘𝑥)(abs ∘ − )(𝐺‘𝑋)) < 𝑑)
132		elbl3 23002	. . . . . . . . . . . . . . . . 17 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ^*) ∧ ((𝐺‘𝑋) ∈ ℂ ∧ (𝐺‘𝑥) ∈ ℂ)) → ((𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ↔ ((𝐺‘𝑥)(abs ∘ − )(𝐺‘𝑋)) < 𝑑))
133	132	biimpar 480	. . . . . . . . . . . . . . . 16 ⊢ (((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑑 ∈ ℝ^*) ∧ ((𝐺‘𝑋) ∈ ℂ ∧ (𝐺‘𝑥) ∈ ℂ)) ∧ ((𝐺‘𝑥)(abs ∘ − )(𝐺‘𝑋)) < 𝑑) → (𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))
134	123, 126, 131, 133	syl21anc 835	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ (abs‘((𝐺‘𝑥) − (𝐺‘𝑋))) < 𝑑) → (𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))
135	119, 134	syldan 593	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ ((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2))) → (𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))
136	135	ex 415	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (((abs‘(ℜ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2) ∧ (abs‘(ℑ‘((𝐺‘𝑥) − (𝐺‘𝑋)))) < (𝑑 / 2)) → (𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)))
137	105, 136	syld 47	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → (𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)))
138		f1ocnv 6627	. . . . . . . . . . . . . . 15 ⊢ (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → ^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ))
139	107, 110, 138	mp2b 10	. . . . . . . . . . . . . 14 ⊢ ^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ)
140		f1ofun 6617	. . . . . . . . . . . . . 14 ⊢ (^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ) → Fun ^◡𝐺)
141	139, 140	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun ^◡𝐺
142		f1odm 6619	. . . . . . . . . . . . . . 15 ⊢ (^◡𝐺:ℂ–1-1-onto→(ℝ × ℝ) → dom ^◡𝐺 = ℂ)
143	139, 142	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom ^◡𝐺 = ℂ
144	116, 143	eleqtrrdi 2924	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝐺‘𝑥) ∈ dom ^◡𝐺)
145		funfvima 6992	. . . . . . . . . . . . 13 ⊢ ((Fun ^◡𝐺 ∧ (𝐺‘𝑥) ∈ dom ^◡𝐺) → ((𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) → (^◡𝐺‘(𝐺‘𝑥)) ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))))
146	141, 144, 145	sylancr 589	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → ((𝐺‘𝑥) ∈ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) → (^◡𝐺‘(𝐺‘𝑥)) ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))))
147	107, 110	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → 𝐺:(ℝ × ℝ)–1-1-onto→ℂ)
148		f1ocnvfv1 7033	. . . . . . . . . . . . . . 15 ⊢ ((𝐺:(ℝ × ℝ)–1-1-onto→ℂ ∧ 𝑥 ∈ (ℝ × ℝ)) → (^◡𝐺‘(𝐺‘𝑥)) = 𝑥)
149	147, 100, 148	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (^◡𝐺‘(𝐺‘𝑥)) = 𝑥)
150	149	eleq1d 2897	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → ((^◡𝐺‘(𝐺‘𝑥)) ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ↔ 𝑥 ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))))
151	150	biimpd 231	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → ((^◡𝐺‘(𝐺‘𝑥)) ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) → 𝑥 ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))))
152	137, 146, 151	3syld 60	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) → (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → 𝑥 ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))))
153	152	imp 409	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ (ℝ × ℝ)) ∧ 𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))) → 𝑥 ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)))
154	95, 153	syl 17	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))) → 𝑥 ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)))
155	154	ex 415	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → (𝑥 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → 𝑥 ∈ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))))
156	155	ssrdv 3973	. . . . . . 7 ⊢ (((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) ∧ 𝑑 ∈ ℝ⁺) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)))
157	156	ralrimiva 3182	. . . . . 6 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∀𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)))
158	103	mpofun 7276	. . . . . . . . . 10 ⊢ Fun 𝐺
159	158	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → Fun 𝐺)
160	13	sselda 3967	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (ℝ × ℝ))
161		f1odm 6619	. . . . . . . . . . 11 ⊢ (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → dom 𝐺 = (ℝ × ℝ))
162	107, 110, 161	mp2b 10	. . . . . . . . . 10 ⊢ dom 𝐺 = (ℝ × ℝ)
163	160, 162	eleqtrrdi 2924	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom 𝐺)
164		simpr 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
165		funfvima 6992	. . . . . . . . . 10 ⊢ ((Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺) → (𝑋 ∈ 𝐴 → (𝐺‘𝑋) ∈ (𝐺 “ 𝐴)))
166	165	imp 409	. . . . . . . . 9 ⊢ (((Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺) ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) ∈ (𝐺 “ 𝐴))
167	159, 163, 164, 166	syl21anc 835	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) ∈ (𝐺 “ 𝐴))
168		hmeoima 22373	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ ((𝐽 ×_t 𝐽)Homeo(TopOpen‘ℂ_fld)) ∧ 𝐴 ∈ (𝐽 ×_t 𝐽)) → (𝐺 “ 𝐴) ∈ (TopOpen‘ℂ_fld))
169	107, 168	mpan 688	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → (𝐺 “ 𝐴) ∈ (TopOpen‘ℂ_fld))
170	106	cnfldtopn 23390	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
171	170	elmopn2 23055	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ∈ (∞Met‘ℂ) → ((𝐺 “ 𝐴) ∈ (TopOpen‘ℂ_fld) ↔ ((𝐺 “ 𝐴) ⊆ ℂ ∧ ∀𝑚 ∈ (𝐺 “ 𝐴)∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))))
172	122, 171	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝐺 “ 𝐴) ∈ (TopOpen‘ℂ_fld) ↔ ((𝐺 “ 𝐴) ⊆ ℂ ∧ ∀𝑚 ∈ (𝐺 “ 𝐴)∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴)))
173	172	simprbi 499	. . . . . . . . . 10 ⊢ ((𝐺 “ 𝐴) ∈ (TopOpen‘ℂ_fld) → ∀𝑚 ∈ (𝐺 “ 𝐴)∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))
174	169, 173	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → ∀𝑚 ∈ (𝐺 “ 𝐴)∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))
175	174	adantr 483	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∀𝑚 ∈ (𝐺 “ 𝐴)∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))
176		oveq1 7163	. . . . . . . . . . 11 ⊢ (𝑚 = (𝐺‘𝑋) → (𝑚(ball‘(abs ∘ − ))𝑑) = ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑))
177	176	sseq1d 3998	. . . . . . . . . 10 ⊢ (𝑚 = (𝐺‘𝑋) → ((𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) ↔ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴)))
178	177	rexbidv 3297	. . . . . . . . 9 ⊢ (𝑚 = (𝐺‘𝑋) → (∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) ↔ ∃𝑑 ∈ ℝ⁺ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴)))
179	178	rspcva 3621	. . . . . . . 8 ⊢ (((𝐺‘𝑋) ∈ (𝐺 “ 𝐴) ∧ ∀𝑚 ∈ (𝐺 “ 𝐴)∃𝑑 ∈ ℝ⁺ (𝑚(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴)) → ∃𝑑 ∈ ℝ⁺ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))
180	167, 175, 179	syl2anc 586	. . . . . . 7 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑑 ∈ ℝ⁺ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴))
181		imass2 5965	. . . . . . . . . 10 ⊢ (((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) → (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (^◡𝐺 “ (𝐺 “ 𝐴)))
182		f1of1 6614	. . . . . . . . . . . . 13 ⊢ (𝐺:(ℝ × ℝ)–1-1-onto→ℂ → 𝐺:(ℝ × ℝ)–1-1→ℂ)
183	107, 110, 182	mp2b 10	. . . . . . . . . . . 12 ⊢ 𝐺:(ℝ × ℝ)–1-1→ℂ
184		f1imacnv 6631	. . . . . . . . . . . 12 ⊢ ((𝐺:(ℝ × ℝ)–1-1→ℂ ∧ 𝐴 ⊆ (ℝ × ℝ)) → (^◡𝐺 “ (𝐺 “ 𝐴)) = 𝐴)
185	183, 13, 184	sylancr 589	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → (^◡𝐺 “ (𝐺 “ 𝐴)) = 𝐴)
186	185	sseq2d 3999	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → ((^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ (^◡𝐺 “ (𝐺 “ 𝐴)) ↔ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
187	181, 186	syl5ib 246	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → (((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) → (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
188	187	reximdv 3273	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐽 ×_t 𝐽) → (∃𝑑 ∈ ℝ⁺ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) → ∃𝑑 ∈ ℝ⁺ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
189	188	adantr 483	. . . . . . 7 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → (∃𝑑 ∈ ℝ⁺ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑) ⊆ (𝐺 “ 𝐴) → ∃𝑑 ∈ ℝ⁺ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
190	180, 189	mpd 15	. . . . . 6 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑑 ∈ ℝ⁺ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴)
191		r19.29 3254	. . . . . 6 ⊢ ((∀𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ ∃𝑑 ∈ ℝ⁺ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴) → ∃𝑑 ∈ ℝ⁺ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
192	157, 190, 191	syl2anc 586	. . . . 5 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑑 ∈ ℝ⁺ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴))
193		sstr 3975	. . . . . 6 ⊢ ((((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴) → ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)
194	193	reximi 3243	. . . . 5 ⊢ (∃𝑑 ∈ ℝ⁺ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ∧ (^◡𝐺 “ ((𝐺‘𝑋)(ball‘(abs ∘ − ))𝑑)) ⊆ 𝐴) → ∃𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)
195	192, 194	syl 17	. . . 4 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)
196		r19.29 3254	. . . 4 ⊢ ((∀𝑑 ∈ ℝ⁺ 𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ∃𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴) → ∃𝑑 ∈ ℝ⁺ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴))
197	70, 195, 196	syl2anc 586	. . 3 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑑 ∈ ℝ⁺ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴))
198		r19.29 3254	. . 3 ⊢ ((∀𝑑 ∈ ℝ⁺ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ ∃𝑑 ∈ ℝ⁺ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)) → ∃𝑑 ∈ ℝ⁺ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)))
199	51, 197, 198	syl2anc 586	. 2 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑑 ∈ ℝ⁺ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)))
200		eleq2 2901	. . . . 5 ⊢ (𝑟 = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → (𝑋 ∈ 𝑟 ↔ 𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2))))))
201		sseq1 3992	. . . . 5 ⊢ (𝑟 = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → (𝑟 ⊆ 𝐴 ↔ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴))
202	200, 201	anbi12d 632	. . . 4 ⊢ (𝑟 = ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) → ((𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴) ↔ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)))
203	202	rspcev 3623	. . 3 ⊢ ((((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)) → ∃𝑟 ∈ 𝐵 (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴))
204	203	rexlimivw 3282	. 2 ⊢ (∃𝑑 ∈ ℝ⁺ (((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∈ 𝐵 ∧ (𝑋 ∈ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ∧ ((((1^st ‘𝑋) − (𝑑 / 2))(,)((1^st ‘𝑋) + (𝑑 / 2))) × (((2^nd ‘𝑋) − (𝑑 / 2))(,)((2^nd ‘𝑋) + (𝑑 / 2)))) ⊆ 𝐴)) → ∃𝑟 ∈ 𝐵 (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴))
205	199, 204	syl 17	1 ⊢ ((𝐴 ∈ (𝐽 ×_t 𝐽) ∧ 𝑋 ∈ 𝐴) → ∃𝑟 ∈ 𝐵 (𝑋 ∈ 𝑟 ∧ 𝑟 ⊆ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 class class class wbr 5066 × cxp 5553 ^◡ccnv 5554 dom cdm 5555 ran crn 5556 “ cima 5558 ∘ ccom 5559 Fun wfun 6349 Fn wfn 6350 ⟶wf 6351 –1-1→wf1 6352 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 1^st c1st 7687 2^nd c2nd 7688 ℂcc 10535 ℝcr 10536 ici 10539 + caddc 10540 · cmul 10542 +∞cpnf 10672 -∞cmnf 10673 ℝ^*cxr 10674 < clt 10675 ≤ cle 10676 − cmin 10870 / cdiv 11297 2c2 11693 ℝ⁺crp 12390 (,)cioo 12739 ℜcre 14456 ℑcim 14457 abscabs 14593 TopOpenctopn 16695 topGenctg 16711 ∞Metcxmet 20530 ballcbl 20532 ℂ_fldccnfld 20545 Topctop 21501 ×_t ctx 22168 Homeochmeo 22361

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cn 21835 df-cnp 21836 df-tx 22170 df-hmeo 22363 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486

This theorem is referenced by: dya2iocnei 31540

W3C validator